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The Perry Preschool Project was created in the early 1960 s by David Weikart in Ypsilanti, Michigan. In this project, 123 African American children were randomly assigned to one of two groups: One group enrolled in the Perry Preschool, and one group did not enroll. Follow-up studies were done for decades. One research question was whether attendance at preschool had an effect on high school graduation. The table shows whether the students graduated from regular high school or not and includes girls only (Schweinhart et al. 2005). $$ \begin{array}{|lcc|} \hline & \text { Preschool } & \text { No Preschool } \\ \hline \text { HS Grad } & 21 & 8 \\ \hline \text { No HS Grad } & 4 & 17 \\ \hline \end{array} $$ a. Find the percentages that graduated for both groups, and compare them descriptively. Does this suggest that preschool was associated with a higher graduation rate? b. Which of the conditions fail so that we cannot use a confidence interval for the difference between proportions?

Step by step solution

01

Calculate the Graduation Percentages

To find out the percentages that graduated from both groups, add up the total number of girls in each group, and divide the number who graduated by the total number. Multiply the result by 100 to get the percentage. For the preschool group: \((21 / (21 + 4)) \times 100 = 84\% \) For the no-preschool group: \((8 / (8 + 17)) \times 100 = 32\%\)

02

Compare the Graduation Percentages

Based on the calculated percentages, we can see a marked difference between the groups. The graduation rate is significantly higher in the group that attended preschool (84%) compared to the group that did not (32%). Thus, this suggests that preschool might have been associated with a higher graduation rate.

03

Check the Conditions for Using a Confidence Interval

In order to use a confidence interval for the difference between proportions, certain conditions must be met: 1. The sample must be random. According to the exercise, this is the case, as children were randomly assigned to each group. 2. The two samples must be independent of each other. This condition is met as well. As the children belong to different groups, they can't be in both groups at the same time. 3. The sample size should be large enough. That is, the number of successes and failures in each group must be at least 10. For the preschool group, this condition is not met as \( 4 < 10\) (4 girls from the preschool group didn't graduate). And for the no-preschool group the number of girls who graduated is 8 which is less than 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Preschool Impact

The Perry Preschool Project sheds light on the potential impact of early education on future academic success. Preschool is often considered an important phase that lays the groundwork for lifelong learning and development.

For example, in this study, 84% of the girls who attended preschool graduated high school, compared to just 32% who did not. This significant difference suggests that early childhood education may provide children with essential skills, knowledge, and an enthusiasm for learning that can have lasting positive effects.

The concept of preschool impact reminds us how foundational experiences shape our ability to succeed later in life. This can include improved cognitive skills, social development, and emotional stability.

Graduation Rates

Graduation rates are a critical metric for evaluating educational outcomes. They represent the percentage of students who successfully complete their schooling within a particular timeframe, typically by obtaining a high school diploma.

By analyzing graduation rates, researchers can infer the effectiveness of specific programs or interventions. In the case of the Perry Preschool Project, we observed stark differences between the preschool and no-preschool groups. A graduation rate of 84% for the preschool group compared to 32% for the non-attendees illustrates the significant role that early education may play in academic trajectories.

Graduation rates give insights into how well educational systems prepare students for the future. They are often used as benchmarks for policy-making, helping to tailor education systems to improve overall student success.

Confidence Intervals

Confidence intervals are a statistical tool that provides a range within which we can expect a population parameter to lie, with a given level of certainty. For instance, they can be used to estimate the difference in graduation rates between two groups.

However, certain conditions must be met to construct meaningful confidence intervals. The sample sizes need to be sufficiently large, and the samples must be randomly selected and independent. If these conditions are not met, the confidence interval might not be reliable.

In the Perry Preschool Project, the sample of non-graduates from preschool was too small, with only 4 students not graduating. This falls short of the requirement for at least 10 individuals in each option (success and failure) per group, indicating that a confidence interval was not feasible for this data.

Random Sampling

Random sampling is a fundamental principle in statistics, ensuring that every individual in a population has an equal chance of being selected for the sample. It aims to create unbiased samples that accurately reflect the population being studied.

The Perry Preschool Project utilized random sampling by randomly assigning children to either participate in the preschool program or not. This random allocation is crucial in reducing selection bias and ensuring that the results - such as the differing graduation rates - are not influenced by pre-existing differences between the groups.

With random sampling, researchers and educators can trust their findings more, allowing valid generalizations to be made to larger populations. However, maintaining truly random samples can be challenging, especially in smaller studies.